Much mental health research uses neurophysiological measurements to describe the way neural activity within and across brain regions is related to behavioral function and dysfunction. One kind of signal, known as spike trains, comes from individual neurons. Other signals, including local field potentials (LFPs), electroencephalography (EEGs), and magnetoencephalography (MEG) are based on activity from large numbers of neurons within specified parts of the brain. With all of these sources of data, scientifically rigorous statistical analysis must accommodate unstable fluctuations, associated with movement or thought, known in statistics as non-stationary. The continuing research program of this grant is to develop methods for analyzing non-stationary neural data. The number of neural signals that can be recorded simultaneously has been increasing rapidly. Because neural network dysfunction is widely considered to be associated with psychopathology, improvements in recording technologies offer exciting opportunities. They also create big statistical challenges due to greatly increased complexity. The research in this grant aims to provide methods for analyzing the ways that network structure may change with particular variables, including those that help characterize behavior, which involves the transmission of neural information at multiple timescales. Fast timescales include oscillations and neural synchrony, which could provide an essential mechanism of neural network information flow and may be a marker that distinguishes normal from diseased states. At slower timescales there is considerable redundancy in the recorded signals, which suggests dimensionality reduction. New methods investigated in this research program can accommodate both faster and slower timescales, and they can also accommodate relationships arising from the spatial configuration of electrodes that record neural signals. These methods are tailored to handle spike trains, LFP, and MEG data, especially as they might arise in experiments related to mental health research. Because a neural spike train is a set of times at which a neuron fired, it is common to consider it to be a point process, which is the statistical model set up to handle sequences of event times. The research supported by this grant concerns development and investigation of statistical techniques involving both multi-dimensional continuous time series (for LFP, EEG, and MEG data) and multi-dimensional point processes (for spike trains).